\subsection*{2012-07-13}

\begin{itemize}
\item for non-relativistic incompressible viscous fluids described by the Navier-Stokes equatios the issue of finding globally regular solutions remains an open challenge.
\item interacting qft equilibriates locally at high enough energy densities
\item the effective theory described in terms of local densities and local fluid velocities. 
\item the equations of fluid dynamics are just the conservation equations + constitutive relations that express the stress-energy tensor $T^{\mu\nu}$ in terms of fluid mechanical variables.
\item systematic procedure to map solution of relativistic fluid dynamics to solutions of gravity in $AdS_5$, with the boundary stress tensor equal to the fluid dynamical stress tensor, and regular away from the black brane singularity. In particular, with a regular horizon.
\item symmetric static solutions to EFE in vacuum in Eddington-Finkelsein coordinates:
\begin{equation}
  ds^2 = 2 dt dr - r^2 f(br) dt^2 + r^2 d\vec x^2
\end{equation}
\item 
These coordinates (and their boosted version to follow) are manifestly regular at the horizon.
Lines of constant $x^\mu$ are radially infalling null geodesics affinely parametrized by $r$.
\item 
Compute the holographic boundary stress-energy tensor, and interpet it as a fluid tensor.  In asymptotically AdS spaces, to compute the tensor we regulate AdS at some finite cutoff $r = r_\epsilon$, and consider the induced metric on this hypersurface, which up to a scale factor involving $r_\epsilon$ is the induced metric $h_{\mu\nu}$. Then compute extrinsic curvature:
\begin{equation}
  K_{\mu \nu} = g_{\mu \rho} \nabla^\rho n_\nu
\end{equation}
Then the stress energy tensor is:
\begin{equation}
T^{\mu\nu} = \lim_{r_\epsilon} \tfrac{r_\epsilon}{16 \pi G_N^{d+1}} \left( K^{\mu \nu} - K h^{\mu\nu} - (d+1) h^{\mu\nu} - \tfrac{1}{d-2}\left( R^\mu \nu - \tfrac{1}{2} R h^{\mu\nu} \right)  \right)
\end{equation}
\item
Turns  out the fluid is stationary. Boost the metric by some constant 4-velocity $u$ ($dx^\mu$ stands for spacetime indicies on the boundary).
\begin{equation}
  ds^2 = -2 u_\mu dx^\mu dr - r^2 f(br) u_\mu u_\nu dx^\mu dx^\nu + r^2 P_{\mu\nu} dx^\mu dx^\nu
\end{equation}
where we have chosen a coordinate-dependent 3-velocity $\beta_i$, and set:
\begin{equation}
  u = \tfrac{(1, \beta_i)}{\sqrt{1-\beta^2}}
\end{equation}
Interpet this velocity as the velocity of the boundary fluid.  
The boundary stress energy tensor is (boundary $d$ spacetime dimensions, corresponding to gravity in $d+1$):
\begin{equation}
  T^{\mu\nu}  =T^d\left(g^{\mu nu} + d u^\mu u^\nu\right)
\end{equation}
\item promote the $u$ to a function of the boundary cooridnates $u = u(x^\mu)$. Similarly, $b = b(x^\mu)$ (recall that $b \sim \tfrac{1}{T}$). The resulting metric is not, in general, a solution to EFE: we have to constrain the form of $u$ and $b$. 
\item 
the resulting metric has interesting features:
\begin{itemize}
  \item everywhere regular
  \item if derivatives of $u$ and $b$ are small, the solution is well-approximated by a black hole (brane).
\end{itemize}

\item To ensure the ``new'' metric solves EFE, we do a derivative expansion: rescale coordinates by a formal parameter $\varepsilon$: $x^\mu 
\rightarrow \varepsilon x^\mu$. Every derivative of $u(x^\mu)$ and $b(x^\mu)$ ``produces'' a factor of $\varepsilon$ by the chain rule. 
So, expand the $u$ and $b$ (and hence the metric) in powers of $\varepsilon$ and set $\varepsilon = 1$ at the end. 
\begin{equation}
  g = g^{(0)} (\beta_i, b) + \varepsilon g^{(1)}\left(\beta_i, b\right) + \varepsilon g^{(2)}\left(\beta_i, b\right) + \mathcal{O}(\varepsilon^3)
\end{equation}

\item Plug the resulting series, order by order, into the EFE and derive constrains on $u$ and $b$. Why can we do this? each derivate gets smaller and smaller by an order of $\tfrac{1}{TL}$, where $L$ is the length scale of variations of $u$ and $b$. The limit $\tfrac{1}{TL} \ll 1$ is also known as the \emph{thermodynamic limit}. By solving the EFE order by order we require that all lower orders cancel \emph{precisely} to 0. {\bf Question:} Is that legal? Couldn't the difference at order $n$, in principle, be still of order $n+1$ (as opposed to 0)? 



\item the Eddington-Finkelstein coordinates provide a clear physical pitcure of the locally equilibrated fluid dynamical domains in the bulk geometry: The boundary domains where local thermal equilibrium is attained extends along ingoing radial null geodesic back into the bulk.

\item Perturbative solutions to the gravitational equations exist only when $u$ and $T$ obey certain equations of motion, which are corrected order by order in $\varepsilon$ expansion. This forces us to correct the $u$ and $T$ themselves, order by order in the expansion.
\begin{equation}
  \beta_i = \beta_i^{(0)} + \varepsilon \beta_i^{(1)} + \mathcal{O}(\varepsilon^2)
\end{equation}
and similarly for the temperature $T$.

\item Constrains derived from EFE turn out to be the local conservation for the fluid stress energy tensor $T$:
\begin{equation}
  \partial_\mu T^{\mu\nu} = 0
\end{equation}
The stress energy tensor receives corrections in higher derivate orders of $u$ (in 3+1 grav)
\begin{equation}
  T^{\mu\nu} = \left(\pi T\right)^4\left(\eta^{\mu\nu}+4 u^\mu u^\nu\right) + O(\nabla u)
\end{equation}
The coefficient of the corrections, the transport coefficients, are explicitly determined by the construction. 

\item note that at nth order of expansion the nth order $\beta_i^{(n)}$ and $b^{(n)}$ do not enter the nth order EFE as the zeroth order terms solve EFE to begin with. 


\item At each order, we end up with $\tfrac{(d+1)(d+2)}{2}$ (15 in $d=3+1$, the spacetime dimension of the boundary) equations from EFE. 
Split them into two categories:
\begin{itemize}
  \item those that teremine metric data, $\tfrac{d(d+1)}{2}$ equations, \emph{dynamical equations}
  \item $d$ \emph{constraint equations}.
\end{itemize}
\item
constraint equations do not involve the unknown metric corrections $g^{(n)}$; they simply constrain the functions $b$ and $u$.
The constraint at order $n$ turns out to be precisely
\begin{equation}
  \partial_\mu T^{\mu\nu}_{(n-1)} = 0
\end{equation}
where $T^{\mu\nu}$ is the boundary stress tensor dual to the metric solution obtained \emph{up to} order $(n-1)$.
This order by order constraining of the terms appearing in $u$ and $b$ is essentially solving hydrodynamics in gradient expansion. If I had a tail I'd wave it. 


\item
the remaining dynamical equations can be used to determine $g^{(n)}$, from which in the next order we compute $T_{\mu\nu}^{(n)}$.

At each order $g^{(n)}$ is built from nth derivatives of $u$ and $b$. There's an overall dependence on the $r$ coordinate as well. 
% The equations with first order dependence on $r$ we dub the ``constraint equations''.
% These are the conservation equations for $T$ (or its constituents build out of derivatives of $b$ and $u$ of the respective order).

\item at zeroth order, the eom are just the conservation of perfect fluid stress-energy tensor:
\begin{equation}
  {T^{\mu\nu}}^{(0)} = \rho u^\mu u^\nu + P \left(g^{\mu\nu} + u^\mu u^\nu\right)
\end{equation}  




\end{itemize}
